Interactive Design of Cubic IPH Spline Curves

Indirect Pythagorean hodographs (IPH) spline curves are a set of curves which have rational Pythagorean hodographs after reparameterization by a fractional quadratic transformation. In this paper, the authors provide an algorithm to interactively design a cubic IPH spline curve from any given control polygon. The method has the same friendly interface and properties as those for B-splines, meanwhile facilitates intuitive and efficient construction of open and closed IPH spline curves. The key idea is to solve the ratios of a set of auxiliary points associated with the edges and then construct a piecewise cubic IPH spline curve which has as high as possible continuity, i.e., the absolute curvature value of the adjacent curve segments are the same. A very interesting observation is that for any open control polygon, a quadratic B-spline curve can have continuous absolute curvature by carefully choosing the knots as the function of the control points.